Abelian tropical covers

Yoav Len; Martin Ulirsch; Dmitry Zakharov

doi:10.1017/S0305004123000518

Abelian tropical covers

Yoav Len^*, Martin Ulirsch, Dmitry Zakharov

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let 𝔄 be a finite abelian group. In this article, we classify harmonic 𝔄-covers of a tropical curve Γ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on Γ. We give a realizability criterion for harmonic 𝔄-covers by patching local monodromy data in an extended homology group on Γ. As an explicit example, we work out the case 𝔄=ℤ/pℤ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.

Original language	English
Pages (from-to)	395 - 416
Number of pages	22
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	176
Issue number	2
Early online date	10 Oct 2023
DOIs	https://doi.org/10.1017/S0305004123000518
Publication status	Published - Mar 2024

Keywords

Coverings
Fundamental group

Access to Document

10.1017/S0305004123000518Licence: CC BY

Len-2023-Abelian-tropical-covers-MPCPS-CCBY
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Final published version, 358 KBLicence: CC BY

1 Active

Tropical Geometry and the moduli space o: Tropical Geometry and the moduli space of Prym varieties
Len, Y. (PI)
EPSRC
1/05/23 → 31/10/25
Project: Standard

1 Preprint

Abelian tropical covers
Len, Y., Ulirsch, M. & Zakharov, D., 2019, (Submitted) arXiv, p. 45.
Research output: Working paper › Preprint

Cite this

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title = "Abelian tropical covers",

abstract = "Let 𝔄 be a finite abelian group. In this article, we classify harmonic 𝔄-covers of a tropical curve Γ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on Γ. We give a realizability criterion for harmonic 𝔄-covers by patching local monodromy data in an extended homology group on Γ. As an explicit example, we work out the case 𝔄=ℤ/pℤ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.",

keywords = "Coverings, Fundamental group",

author = "Yoav Len and Martin Ulirsch and Dmitry Zakharov",

note = "Funding: This project has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme under the Marie–Sk{\l}odowska–Curie Grant Agreement No. 793039. M.U. has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Sachbeihilfe From Riemann surfaces to tropical curves (and back again), project number 456557832, and from the LOEWE grant Uniformized Structures in Algebra and Geometry. Y.L. has received funding from the EPSRC New Investigator Award (grant number EP/X002004/1).",

year = "2024",

month = mar,

doi = "10.1017/S0305004123000518",

language = "English",

volume = "176",

pages = "395 -- 416",

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AU - Len, Yoav

AU - Ulirsch, Martin

AU - Zakharov, Dmitry

N1 - Funding: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie–Skłodowska–Curie Grant Agreement No. 793039. M.U. has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Sachbeihilfe From Riemann surfaces to tropical curves (and back again), project number 456557832, and from the LOEWE grant Uniformized Structures in Algebra and Geometry. Y.L. has received funding from the EPSRC New Investigator Award (grant number EP/X002004/1).

PY - 2024/3

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N2 - Let 𝔄 be a finite abelian group. In this article, we classify harmonic 𝔄-covers of a tropical curve Γ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on Γ. We give a realizability criterion for harmonic 𝔄-covers by patching local monodromy data in an extended homology group on Γ. As an explicit example, we work out the case 𝔄=ℤ/pℤ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.

AB - Let 𝔄 be a finite abelian group. In this article, we classify harmonic 𝔄-covers of a tropical curve Γ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on Γ. We give a realizability criterion for harmonic 𝔄-covers by patching local monodromy data in an extended homology group on Γ. As an explicit example, we work out the case 𝔄=ℤ/pℤ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.

KW - Coverings

KW - Fundamental group

U2 - 10.1017/S0305004123000518

DO - 10.1017/S0305004123000518

M3 - Article

SN - 0305-0041

VL - 176

SP - 395

EP - 416

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 2

ER -

Abelian tropical covers

Abstract

Keywords

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Projects

Tropical Geometry and the moduli space o: Tropical Geometry and the moduli space of Prym varieties

Research output

Abelian tropical covers

Cite this